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Finding, minimizing, and counting weighted subgraphs
 In Proceedings of the FourtyFirst Annual ACM Symposium on the Theory of Computing
, 2009
"... For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both nodeweighted and edgeweighted graphs. Our results include: • The number of cop ..."
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Cited by 14 (2 self)
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For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both nodeweighted and edgeweighted graphs. Our results include: • The number of copies of an H with an independent set of size s can be computed exactly in O ∗ (2 s n k−s+3) time. A minimum weight copy of such an H (with arbitrary real weights on nodes and edges) can be found in O(4 s+o(s) n k−s+3) time. (The O ∗ notation omits poly(k) factors.) These algorithms rely on fast algorithms for computing the permanent of a k × n matrix, over rings and semirings. • The number of copies of any H having minimum (or maximum) nodeweight (with arbitrary real weights on nodes) can be found in O(n ωk/3 + n 2k/3+o(1) ) time, where ω < 2.4 is the matrix multiplication exponent and k is divisible by 3. Similar results hold for other values of k. Also, the number of copies having exactly a prescribed weight can be found within this time. These algorithms extend the technique of Czumaj and Lingas (SODA 2007) and give a new (algorithmic) application of multiparty communication complexity. • Finding an edgeweighted triangle of weight exactly 0 in general graphs requires Ω(n 2.5−ε) time for all ε> 0, unless the 3SUM problem on N numbers can be solved in O(N 2−ε) time. This suggests that the edgeweighted problem is much harder than its nodeweighted version. 1
Subcubic Equivalences Between Path, Matrix, and Triangle Problems ∗
"... We say an algorithm on n × n matrices with entries in [−M,M] (or nnode graphs with edge weights from [−M,M]) is truly subcubic if it runs in O(n 3−δ · poly(log M)) time for some δ> 0. We define a notion of subcubic reducibility, and show that many important problems on graphs and matrices solvable ..."
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Cited by 10 (5 self)
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We say an algorithm on n × n matrices with entries in [−M,M] (or nnode graphs with edge weights from [−M,M]) is truly subcubic if it runs in O(n 3−δ · poly(log M)) time for some δ> 0. We define a notion of subcubic reducibility, and show that many important problems on graphs and matrices solvable in O(n 3) time are equivalent under subcubic reductions. Namely, the following weighted problems either all have truly subcubic algorithms, or none of them do: • The allpairs shortest paths problem on weighted digraphs (APSP). • Detecting if a weighted graph has a triangle of negative total edge weight. • Listing up to n 2.99 negative triangles in an edgeweighted graph. • Finding a minimum weight cycle in a graph of nonnegative edge weights. • The replacement paths problem on weighted digraphs. • Finding the second shortest simple path between two nodes in a weighted digraph. • Checking whether a given matrix defines a metric. • Verifying the correctness of a matrix product over the (min,+)semiring. Therefore, if APSP cannot be solved in n 3−ε time for any ε> 0, then many other problems also
Computing the Discrete Fréchet Distance in Subquadratic Time ∗
, 2012
"... The Fréchet distance is a similarity measure between two curves ..."
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Cited by 2 (1 self)
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The Fréchet distance is a similarity measure between two curves
Necklaces, Convolutions, and X + Y
"... Abstract. We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the ℓp norm of the vector of distances between pairs of beads from opposite nec ..."
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Abstract. We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the ℓp norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p = 1, p = 2, and p = ∞. For p = 2, we reduce the problem to standard convolution, while for p = ∞ and p = 1, we reduce the problem to (min, +) convolution and (median, +) convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X + Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X + Y matrix. All of our algorithms run in o(n 2) time, whereas the obvious algorithms for these problems run in Θ(n 2) time. 1
The Complexity of Linear Dependence Problems in Vector Spaces
"... We study extensions of the natural kSUM problem to vector spaces over finite fields. Given a subset S ⊆ F n q of size r ≤ q n, an integer k, 2 ≤ k ≤ n, and a vector v ∈ (F k q \ {0}) k, we define the TargetSum problem to be the problem of finding k elements xi1,..., xik ∈ S for which ∑k vjxij j=1 = ..."
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We study extensions of the natural kSUM problem to vector spaces over finite fields. Given a subset S ⊆ F n q of size r ≤ q n, an integer k, 2 ≤ k ≤ n, and a vector v ∈ (F k q \ {0}) k, we define the TargetSum problem to be the problem of finding k elements xi1,..., xik ∈ S for which ∑k vjxij j=1 = z, where z may either be an input or a fixed vector. We also study a variant of this, where instead of finding xi1,..., xik ∈ S for which ∑k vjxij = z, we require that z be in j=1 span(xi1,..., xik), which we call the (k, r)LinDependenceq problem. These problems are natural generalizations of wellstudied problems that occur in coding theory and property testing. Indeed, the (k, r)LinDependenceq problem is just the Maximum Likelihood Decoding problem. Also, in the TargetSum problem, if instead of general z we require z = 0n, then this is the Weight Distribution problem. In property testing, these problems have been implicitly studied in the context of testing linearinvariant properties. We give nearly optimal bounds for TargetSum and (k, r)LinDependenceq for every r, k, and constant q. Namely, assuming 3SAT requires exponential time, we show that any algorithm for these problems must run in min(2Θ(n) , rΘ(k) ) time. Moreover, we give deterministic upper bounds that match this complexity, up to small factors. Our lower bound strengthens and simplifies an earlier min(2Θ(n) , rΩ(k1/4)) lower bound for both the Maximum Likelihood Decoding and Weight Distribution problems. We also give upper and lower bounds for variants of these problems, e.g., for the problem for which we must find xi1,..., xik for which z ∈ span(xi1,..., xik) but z is not spanned by any proper subset of these vectors, and for the counting version of this problem. Part of this work was done while the author was an intern at IBM Almaden.
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"... 1 Introduction How should we rotate two necklaces, each with n beads at different locations, tobest align the beads? More precisely, each necklace is represented by a set of ..."
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1 Introduction How should we rotate two necklaces, each with n beads at different locations, tobest align the beads? More precisely, each necklace is represented by a set of
Finding, Minimizing, and Counting Weighted Subgraphs ∗ [Extended Abstract]
"... For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both nodeweighted and edgeweighted graphs. Our results include: • The number of co ..."
Abstract
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For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both nodeweighted and edgeweighted graphs. Our results include: • The number of copies of an H with an independent set of size s can be computed exactly in O ∗ (2 s n k−s+3) time. A minimum weight copy of such an H (with arbitrary real weights on nodes and edges) can be found in O(4 s+o(s) n k−s+3) time. (The O ∗ notation omits poly(k) factors.) These algorithms rely on fast algorithms for computing the permanent of a k × n matrix, over rings and semirings. • The number of copies of any H having minimum (or
Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time ∗
, 2008
"... Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linearspace data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unitcost RAM with word size w. Thisisthe first result to beat the standard Θ(lg n) bou ..."
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Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linearspace data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unitcost RAM with word size w. Thisisthe first result to beat the standard Θ(lg n) bound for infinite precision models. As a consequence, we obtain the first o(n lg n) (randomized) algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word RAM, including: constructing the convex hull of a threedimensional point set, computing the Voronoi diagram or the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. Higherdimensional extensions and applications are also discussed. Though computational geometry with bounded precision input has been investigated for a long time, improvements have been limited largely to problems of an orthogonal flavor. Our results surpass this longstanding limitation, answering, for example, a question of Willard (SODA’92). Key words. Computational geometry, wordRAM algorithms, data structures, sorting, searching, convex hulls, Voronoi diagrams, segment intersection AMS subject classifications. 68Q25, 68P05, 68U05 Abbreviated title. Point location in sublogarithmic time
Scalable Computation of Acyclic Joins (Extended Abstract)
, 2006
"... The join operation of relational algebra is a cornerstone of relational database systems. Computing the join of several relations is NPhard in general, whereas special (and typical) cases are tractable. This paper considers joins having an acyclic join graph, for which current methods initially app ..."
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The join operation of relational algebra is a cornerstone of relational database systems. Computing the join of several relations is NPhard in general, whereas special (and typical) cases are tractable. This paper considers joins having an acyclic join graph, for which current methods initially apply a full reducer to efficiently eliminate tuples that will not contribute to the result of the join. From a worstcase perspective, previous algorithms for computing an acyclic join of k fully reduced relations, occupying a total of n ≥ k blocks on disk, use Ω((n + z)k) I/Os,wherez is the size of thejoinresultinblocks. In this paper we show how to compute the join in a time bound that is within a constant factor of the cost of running a full reducer plus sorting the output. For a broad class of acyclic join graphs this is O(sort(n + z)) I/Os, removing the dependence on k from previous bounds. Traditional methods decompose the join into a number of binary joins, which are then carried out one by one. Departing from this approach, our technique is based on computing the size of certain subsets of the result, and using these sizes to compute the location(s) of each data item in the result. Finally, as an initial study of cyclic joins in the I/O model, we show how to compute a join whose join graph is a 3cycle, in O(n 2 /m +sort(n + z)) I/Os, where m is the number of blocks in internal memory.